Spherical varieties, L-functions, and crystal bases

Spherical varieties, L-functions, and crystal bases

Jonathan Wang (joint w/ Yiannis Sakellaridis)

MIT

MIT Lie Groups Seminar, September 23, 2020

Notes available at:http://jonathanpwang.com/notes/sphL_talk_notes.pdf

Jonathan Wang (MIT) Spherical varieties, L-functions, crystals September 23, 2020 1 / 19

Outline

1 What is a spherical variety?

2 Function-theoretic results

3 Geometry

F = Fq((t)), O = Fq[[t]]

k = Fq

G connected split reductive group /Fq


k E

What is a spherical variety?

Definition

A G -variety X/Fqis called spherical if Xk is normal and has an open dense

orbit of Bk ⇢ Gk after base change to k

Think of this as a finiteness condition (good combinatorics)Examples:

Toric varieties G = T

Symmetric spaces K\GGroup X = G 0 G 0 ⇥ G 0 = G



Definition



Think of this as a finiteness condition (good combinatorics)

Examples:




EA X has finitelymany B orbits


Definition



Think of this as a finiteness condition (good combinatorics)Examples:




Why are they relevant?

Conjecture (Sakellaridis, Sakellaridis–Venkatesh)

For any a�ne spherical G -variety X (*),and an irreducible unitary G (F )-representation ⇡, there is an “integral”

|PX |2⇡ : ⇡ ⌦ ⇡ ! C

involving the IC function of X (O) such that

1 |PX |2⇡ 6= 0 determines a functorial lifting of ⇡ to � 2 Irr(GX (F ))corresponding to a map GX (C) ! G (C),

2 there should exist a GX -representation

⇢X : GX (C) ! GL(VX )

such that |PX |2⇡“=”L(�, ⇢X , s0) for a special value s0.





|PX |2⇡ : ⇡ ⌦ ⇡ ! C




⇢X : GX (C) ! GL(VX )






|PX |2⇡ : ⇡ ⌦ ⇡ ! C




⇢X : GX (C) ! GL(VX )






|PX |2⇡ : ⇡ ⌦ ⇡ ! C




⇢X : GX (C) ! GL(VX )



WEft

Excel Eid




|PX |2⇡ : ⇡ ⌦ ⇡ ! C




⇢X : GX (C) ! GL(VX )



Later Ex IO

m

oG F

Some history on GX

Goal: a map GX ! G with finite kernel

TX is easy to define

Little Weyl group WX and spherical root systemSymmetric variety: Cartan ’27Spherical variety: Luna–Vust ’83, Brion ’90; reflection group offundamental domainIrreducible G -variety: Knop ’90, ’93, ’94; moment map, invariantdi↵erential operators

Gaitsgory–Nadler ’06: define subgroup GGN

X⇢ G using Tannakian

formalism

Sakellaridis–Venkatesh ’12: normalized root system, define GX ! Gcombinatorially with image GGN

Xunder assumptions about GN

Knop–Schalke ’17: define GX ! G combinatorially unconditionally


Some history on GX






formalism





Some history on GX



Little Weyl group WX and spherical root systemSymmetric variety: Cartan ’27

Spherical variety: Luna–Vust ’83, Brion ’90; reflection group offundamental domainIrreducible G -variety: Knop ’90, ’93, ’94; moment map, invariantdi↵erential operators



formalism





Some history on GX



Little Weyl group WX and spherical root systemSymmetric variety: Cartan ’27Spherical variety: Luna–Vust ’83, Brion ’90; reflection group offundamental domain

Irreducible G -variety: Knop ’90, ’93, ’94; moment map, invariantdi↵erential operators



formalism





Some history on GX






formalism





TH g DWG

Some history on GX






formalism





Some history on GX






formalism





X G GX VX

Usual Langlands G 0 G 0 ⇥ G 0 G 0 g0

Whittaker normal-ization

(N, )\G G 0

Hecke Gm\PGL2 G = SL2 T ⇤std

Rankin–Selberg,Jacquet–Piatetski-Shapiro–Shalika

H\GLn ⇥ GLn =GLn ⇥ An

G T ⇤(std⌦std)

Gan–Gross–Prasad SO2n\SO2n+1⇥SO2nG = SO2n ⇥Sp2n

std⌦ std


w

X G GX VX



(N, )\G G 0




G T ⇤(std⌦std)


std⌦ std


CNN.MG coca Nitw

X G GX VX



(N, )\G G 0




G T ⇤(std⌦std)


std⌦ std


f U U U

X G GX VX



(N, )\G G 0




G T ⇤(std⌦std)


std⌦ std


X gH diagonal GLnxGLn Gmirabolic

X G GX VX



(N, )\G G 0




G T ⇤(std⌦std)


std⌦ std


mm

Example (Sakellaridis)

G = GL⇥n

2 ⇥Gm, H =

( a x1

1

!⇥

a x21

!⇥ · · ·⇥

a xn

1

!⇥ a

�� x1 + · · ·+ xn = 0

)

X = H\G

GX = G = GL⇥n

2 ⇥Gm

VX = T ⇤(std⌦n

2 ⌦ std1).

To find new interesting examples, need to consider singular X 6= H\G .

Theorem (Luna, Richardson)

H\G is a�ne if and only if H is reductive



G = GL⇥n

2 ⇥Gm, H =

( a x1

1

!⇥

a x21

!⇥ · · ·⇥

a xn

1

!⇥ a

�� x1 + · · ·+ xn = 0

)

X = H\G

GX = G = GL⇥n

2 ⇥Gm

VX = T ⇤(std⌦n

2 ⌦ std1).






G = GL⇥n

2 ⇥Gm, H =

( a x1

1

!⇥

a x21

!⇥ · · ·⇥

a xn

1

!⇥ a

�� x1 + · · ·+ xn = 0

)

X = H\G

GX = G = GL⇥n

2 ⇥Gm

VX = T ⇤(std⌦n

2 ⌦ std1).





GX = G

For this talk, assume GX = G (and X has no type N roots). [‘N’ is fornormalizer]

Equivalent to:

(Base change to k)

X has open B-orbit X � ⇠= B

X �P↵/R(P↵) ⇠= Gm\PGL2 for every simple ↵, P↵ � B


Avoid OMGLn

GX = G

For this talk, assume GX = G (and X has no type N roots). [‘N’ is fornormalizer]

Equivalent to:

(Base change to k)

X has open B-orbit X � ⇠= B

X �P↵/R(P↵) ⇠= Gm\PGL2 for every simple ↵, P↵ � B


Xo c Xo Eq

our

PHRCpo PGL2

Sakellaridis–Venkatesh a la Bernstein

Definition

Fix x0 2 X �(Fq) in open B-orbit. Define the X -Radon transform

⇡! : C1c (X (F ))G(O) ! C1(N(F )\G (F ))G(O)

by

⇡!�(g) :=

Z

N(F )�(x0ng)dn, g 2 G (F )

⇡!� is a function on N(F )\G (F )/G (O) = T (F )/T (O) = ⇤.

Related:

spherical functions (unramified Hecke eigenfunction) on X (F )

unramified Plancherel measure on X (F )


asymptotics

run


Definition


⇡! : C1c (X (F ))G(O) ! C1(N(F )\G (F ))G(O)

by

⇡!�(g) :=

Z

N(F )�(x0ng)dn, g 2 G (F )


Related:




um


Definition


⇡! : C1c (X (F ))G(O) ! C1(N(F )\G (F ))G(O)

by

⇡!�(g) :=

Z

N(F )�(x0ng)dn, g 2 G (F )


Related:





Definition


⇡! : C1c (X (F ))G(O) ! C1(N(F )\G (F ))G(O)

by

⇡!�(g) :=

Z

N(F )�(x0ng)dn, g 2 G (F )


Related:




Conjecture 1 (Sakellaridis–Venkatesh)

Assume GX = G and X has no type N roots.

There exists a symplectic VX 2 Rep(G ) with a T polarizationVX = V+

X� (V+

X)⇤ such that

⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

2 Fn(⇤)

where e � is the indicator function of �, e �eµ = e �+µ

Mellin transform of right hand side gives

� 2 T (C) 7!L(�,V+

X, 12)

L(�, n, 1), this is “half” of

L(�,VX ,12)

L(�, g/t, 1)



Assume GX = G and X has no type N roots.There exists a symplectic VX 2 Rep(G ) with a T polarizationVX = V+

X� (V+

X)⇤ such that

⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

2 Fn(⇤)



� 2 T (C) 7!L(�,V+

X, 12)


L(�,VX ,12)

L(�, g/t, 1)




X� (V+

X)⇤ such that

⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

2 Fn(⇤)



� 2 T (C) 7!L(�,V+

X, 12)


L(�,VX ,12)

L(�, g/t, 1)


Tie Eo Cq e'In



X� (V+

X)⇤ such that

⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

2 Fn(⇤)



� 2 T (C) 7!L(�,V+

X, 12)


L(�,VX ,12)

L(�, g/t, 1)


IxEx Vy of

YINKA

et Ek isnormalizat

Warning Vf never E repe g Heche atd VI Glo lo 1

Previous work

Conjecture 1 (possibly with GX 6= G ) was proved in the following cases:

Sakellaridis (’08, ’13):X = H\G and H is reductive (i↵ H\G is a�ne), no assumption on GX

doesn’t consider X ) H\GBraverman–Finkelberg–Gaitsgory–Mirkovic [BFGM] ’02:

X = N�\G , GX = T , VX = n

Bouthier–Ngo–Sakellaridis [BNS] ’16:X toric variety, G = T , GX = T , weights of VX correspond to latticegenerators of a coneX � G 0 is L-monoid, G = G 0 ⇥ G 0, GX = G 0, VX = g0 � V �


Previous work



doesn’t consider X ) H\G

Braverman–Finkelberg–Gaitsgory–Mirkovic [BFGM] ’02:

X = N�\G , GX = T , VX = n



functiontheoretic smooth

Previous work




X = N�\G , GX = T , VX = n



m

Previous work




X = N�\G , GX = T , VX = n

Bouthier–Ngo–Sakellaridis [BNS] ’16:X toric variety, G = T , GX = T , weights of VX correspond to latticegenerators of a cone

X � G 0 is L-monoid, G = G 0 ⇥ G 0, GX = G 0, VX = g0 � V �


Previous work




X = N�\G , GX = T , VX = n



function theoretic

O

I

Theorem (Sakellaridis–W)

Assume X a�ne spherical, GX = G and X has no type N roots. Then

⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X

2 Rep(T ) such that:

1 V 0X:= V+

X� (V+

X)⇤ has action of (SL2)↵ for every simple root ↵

We do not check Serre relations

2 Assuming V 0Xsatisfies Serre relations (so it is a G -rep), we determine

its highest weights with multiplicities (in terms of X )

(2) gives recipe for conjectural VX in terms of XIf VX is minuscule, then VX = V 0

X.

Proposition

If X = H\G with H reductive, then VX is minuscule.




⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X


1 V 0X:= V+

X� (V+






X.

Proposition



m



⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X


1 V 0X:= V+

X� (V+






X.

Proposition





⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X


1 V 0X:= V+

X� (V+






X.

Proposition





⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X


1 V 0X:= V+

X� (V+





(2) gives recipe for conjectural VX in terms of X

If VX is minuscule, then VX = V 0X.

Proposition



e



⇡!�ICX (O)=

Q↵2�+

G

(1� q�1e↵)Q

�2wt(V+X)(1� q�

12 e �)

for some V+X


1 V 0X:= V+

X� (V+






X.

Proposition



reduce tocases G ss runk2

Enter geometry

Base change to k = Fq (or k = C)XO(k) = X (k[[t]])

XF(k) = X (k((t)))

Problem: XO is an infinite type scheme

Bouthier–Ngo–Sakellaridis: IC function still makes sense byGrinberg–Kazhdan theorem


Enter geometry


XF(k) = X (k((t)))




no perversesheaves

Enter geometry


XF(k) = X (k((t)))




char 0

Zastava space

Drinfeld’s proof of Grinberg–Kazhdan theorem gives an explicit model forXO:

Definition

Let C be a smooth curve over k . Define

Y = Mapsgen(C ,X/B � X �/B)

Following Finkelberg–Mirkovic, we call this the Zastava space of X .

Fact: Y is an infinite disjoint union of finite type schemes.

Y

A

\{⇤-valued divisors on C}

⇡


X B

Zastava space


Definition





Y

A


⇡


f c KBfun x B p

Zastava space


Definition





Y

A


⇡


X

Zastava space


Definition





Y

A


⇡


Zastava space


Definition





Y

A


⇡


y C 7 113u U

O f C Erik pt

a II ri

Define the central fiber Y� = ⇡�1(� · v) for a single point v 2 C (k).

Y Y�

A � · v

Graded factorization property

The fiber ⇡�1(�1v1 + �2v2) for distinct v1, v2 is isomorphic to Y�1 ⇥ Y�2 .

Upshot

⇡!�ICXO(t �) = tr(Fr, (⇡!ICY)|⇤�·v )


it


Y Y�

A � · v



Upshot

⇡!�ICXO(t �) = tr(Fr, (⇡!ICY)|⇤�·v )


m


Y Y�

A � · v



Upshot

⇡!�ICXO(t �) = tr(Fr, (⇡!ICY)|⇤�·v )


Semi-small map

Can compactify ⇡ to proper map ⇡ : Y ! A.


Under previous assumptions, ⇡ : Y ! A is stratified semi-small. Inparticular, ⇡!ICY is perverse.

If Y is smooth, then semi-smallness amounts to the inequality

dimY� crit(�)

Decomposition theorem + factorization property imply

Euler product

tr(Fr, (⇡!ICY)|⇤?·v ) =

1Q

�2B+(1� q�12 e �)

B+ = irred. components of Y�of dim = crit(�) as � varies


Special EyEG

w

Semi-small map





dimY� crit(�)


Euler product

tr(Fr, (⇡!ICY)|⇤?·v ) =

1Q

�2B+(1� q�12 e �)



s

Semi-small map





dimY� crit(�)


Euler product

tr(Fr, (⇡!ICY)|⇤?·v ) =

1Q

�2B+(1� q�12 e �)



F vs Y

Semi-small map





dimY� crit(�)


Euler product

tr(Fr, (⇡!ICY)|⇤?·v ) =

1Q

�2B+(1� q�12 e �)



relevant stratum supportedatv


Define V+X

to have basis B+

Formally set B = B+ t (B+)⇤, so (B+)⇤ is a basis of (V+X)⇤


B has the structure of a (Kashiwara) crystal, i.e., graph with weightedvertices and edges $ raising/lowering operators e↵, f↵

Crystal basis is the (Lusztig) canonical basis at q = 0 of a f.d.Uq(g)-module.

f.d. G -representation crystal basis 2 {crystals}



Define V+X

to have basis B+








Define V+X

to have basis B+







In rn

I 5


Define V+X

to have basis B+







F Fso


Define V+X

to have basis B+








Define V+X

to have basis B+







q l 9 0

Conjecture 2

B is the crystal basis for a finite dimensional G -representation VX .

Conjecture 2 implies Conjecture 1 (B $ V 0X).

Conjecture 2 resembles geometric constructions of crystal bases byLusztig, Braverman–Gaitsgory, Kamnitzer involving irreduciblecomponents of GrG

Y�,Y� ⇢ GrG


Conjecture 2

B is the crystal basis for a finite dimensional G -representation VX .

Conjecture 2 implies Conjecture 1 (B $ V 0X).

Conjecture 2 resembles geometric constructions of crystal bases byLusztig, Braverman–Gaitsgory, Kamnitzer involving irreduciblecomponents of GrG

Y�,Y� ⇢ GrG


S N.pt e Gra X HIr

Y as qq.atd9

Y xc5X Us fetafeet

determined

ngt a nyxa xoria


Documents

Spherical varieties, L-functions, and crystal bases